# How do you find the asymptotes for f(x)= x/(x(x-2))?

Mar 28, 2016

Vertical asymptotes, horizontal asymptotes and "holes" can be found in this function.

#### Explanation:

First, cancel the factor of x from both the numerator and denominator of the expression:
$f \left(x\right) = \frac{1}{x - 2}$

That factor being removed, causes a "hole" in the graph. This is sometimes called a removable discontinuity. So, at $x \ne 0$, there is a "hole".

To find a vertical asymptote, set the denominator equal to 0 and solve:
$x - 2 = 0$ so x = 2 is the vertical asymptote.

To find a horizontal asymptote, notice that the degree of x in the denominator is higher than the degree of x in the numerator. That is, ${x}^{0}$ in the numerator and ${x}^{1}$ in the denominator. Therefore, large positive or negative values of x will cause the expression to approach 0. That means that a horizontal asymptote will be y = 0.

Still not sure about this? Substitute in large positive and negative values for x like 1000000 or -1000000...Your output values will be very, very small!